Editing Waveplate (section)

=== Quarter-wave plate ===
[[File:Circular.Polarization.Circularly.Polarized.Light And.Linearly.Polarized.Light.Comparison.svg|thumb|right|Two waves differing by a quarter-phase shift for one axis]]
[[File:Circular.Polarization.Circularly.Polarized.Light Circular.Polarizer Creating.Left.Handed.Helix.View.svg|thumb|right|Creating circular polarization using a quarter-wave plate and a polarizing filter]]
For a quarter-wave plate, the relationship between ''L'', Δ''n'', and λ<sub>0</sub> is chosen so that the phase shift between polarization components is Γ&nbsp;= π/2. Now suppose a linearly polarized wave is incident on the crystal. This wave can be written as
:<math>(E_f \mathbf{\hat f} + E_s \mathbf{\hat s})\mathrm{e}^{i(kz-\omega t)},</math>
where the ''f'' and ''s'' axes are the quarter-wave plate's fast and slow axes, respectively, the wave propagates along the ''z'' axis, and ''E<sub>f</sub>'' and ''E<sub>s</sub>'' are real. The effect of the quarter-wave plate is to introduce a phase shift term e<sup>''i''Γ</sup>&nbsp;=e<sup>''i''π/2</sup>&nbsp;= ''i'' between the ''f'' and ''s'' components of the wave, so that upon exiting the crystal the wave is now given by
:<math>(E_f \mathbf{\hat f} + i E_s \mathbf{\hat s})\mathrm{e}^{i(kz-\omega t)}.</math>
The wave is now elliptically polarized.

If the axis of polarization of the incident wave is chosen so that it makes a 45° with the fast and slow axes of the waveplate, then ''E<sub>f</sub>''&nbsp;= ''E<sub>s</sub>''&nbsp;≡ ''E'', and the resulting wave upon exiting the waveplate is
:<math>E(\mathbf{\hat f}+i\mathbf{\hat s})\mathrm{e}^{i(kz-\omega t)},</math>
and the wave is circularly polarized.

If the axis of polarization of the incident wave is chosen so that it makes a 0° with the fast or slow axes of the waveplate, then the polarization will not change, so remains linear. If the angle is in between 0° and 45° the resulting wave has an elliptical polarization.

A circulating polarization can be visualized as the sum of two linear polarizations with a phase difference of 90°. The output depends on the polarization of the input. Suppose polarization axes x and y parallel with the slow and fast axis of the waveplate:
[[File:Polaryzacja kołowa.gif|alt=Composition of two linearly polarized waves, phase shifted by π/2|thumb|Composition of two linearly polarized waves, phase shifted by π/2]]
[[File:Quarter wave plate polarizaton.gif]]

The polarization of the incoming photon (or beam) can be resolved as two polarizations on the x and y axis. If the input polarization is parallel to the fast or slow axis, then there is no polarization of the other axis, so the output polarization is the same as the input (only the phase more or less delayed). If the input polarization is 45° to the fast and slow axis, the polarization on those axes are equal. But the phase of the output of the slow axis will be delayed 90° with the output of the fast axis. If not the amplitude but both sine values are displayed, then x and y combined will describe a circle. With other angles than 0° or 45° the values in fast and slow axis will differ and their resultant output will describe an ellipse.